At time \(t = 0\), a particle \(P\) of mass 0.4 kg is at the origin \(O\) moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the \(x\)-axis in the positive \(x\) direction. At time \(t\) seconds, \(t \geqslant 0\), the resultant force acting on \(P\) has magnitude \(\frac { 4 } { ( t + 5 ) ^ { 2 } } \mathrm {~N}\) and is directed away from \(O\).
Show that the speed of \(P\) cannot exceed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
The particle passes through the point \(A\) when \(t = 2\) and passes through the point \(B\) when \(t = 7\)
Find the distance \(A B\).
Find the gain in kinetic energy of \(P\) as it moves from \(A\) to \(B\).